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A non-commutative, analytic version of Hilbert's 17-th problem in type II$_1$ von Neumann algebras

Florin Radulescu
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Abstract

We prove a non-commutative version of the Hilbert's 17th problem, giving a characterization of the class of non-commutative polynomials in n-undeterminates that have positive trace when evaluated in n-selfadjoint elements in arbitrary II1 von Neumann algebra. As a corollary we prove that Connes's embedding conjecture is equivalent to a statement that can be formulated entirely in the context of finite matrices.

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Florin Radulescu. 2005-02-12. A non-commutative, analytic version of Hilbert's 17-th problem in type II$_1$ von Neumann algebras. https://arxiv.org/abs/math/0404458

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